Group Homomorphisms Review

# Group Homomorphisms Review

We will now review some of the recent material regarding group homomorphisms.

- On the
**Group Homomorphisms**a**Homomorphism**from $G$ to $H$ is a function $f : G \to H$ which has the following property that for all $x, y \in G$:

\begin{align} \quad f(x \cdot y) = f(x) * f(y) \end{align}

- We then looked at some important results regarding group homomorphisms on the
**Basic Theorems Regarding Group Homomorphisms**page which are summarized below:

Theorem |
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(a) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $e$ is the identity in $G$ then $f(e)$ is the identity in $H$. |

(b) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $x \in G$ has inverse $x^{-1} \in G$ then $f(x) \in H$ has inverse $f(x^{-1}) \in H$. |

(c) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $H \subseteq G$ is a subgroup of $G$ then $f(H) \subseteq H$ is a subgroup of $H$. |

(d) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and if $H \subseteq H$ is a subgroup of $H$ and $f^{-1}(H) \neq \emptyset$ then $f^{-1}(H)$ is a subgroup of $G$. |

- On
**The Kernel of a Group Homomorphism**page we said that if $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ where $e_2$ is the identity of $H$ then the**Kernel of $f$**is defined as:

\begin{align} \quad \mathrm{ker} (f) = \{ x \in G : f(x) = e_2 \} \end{align}

- That is, $\mathrm{ker} (f)$ is the set of all elements in $G$ that are mapped to the identity element in $H$.

- We noted that $\mathrm{ker} (f)$ is a subgroup of $G$.

- We then proved an important result:

Theorem |
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(a) If $(G, \cdot)$ and $(H, *)$ are homomorphic with homomorphism $f : G \to H$ and with identities $e_1$ and $e_2$ respectively, then $f$ is injective if and only if $\ker (f) = \{ e_1 \}$. |